6.1 Areas Between Curves/19

19)

${\begin{aligned}&\color {red}\mathbf {x=4+y^{2}} &\color {royalblue}\mathbf {x=2y^{2}} \\\end{aligned}}$

${\begin{aligned}4+y^{2}&=2y^{2}\\4&=y^{2}\\y&={\sqrt {4}}\\y&=\pm 2\\[2ex]\int _{-2}^{2}[(4+y^{2})-(2y^{2})]dy\\=\int _{-2}^{2}[4-y^{2}]dy\\[2ex]=\left[4y-{\frac {y^{3}}{3}}\right]{\Bigg |}_{-2}^{2}\\[2ex]=4(2)-{\frac {(2)^{3}}{3}}-(4(-2)-{\frac {(-2)^{3}}{3}})\\[2ex]=8-{\frac {8}{3}}-(-8+{\frac {8}{3}})\\[2ex]=16-{\frac {16}{3}}\\[2ex]={\frac {48}{3}}-{\frac {16}{3}}\\[2ex]={\frac {32}{3}}\\\end{aligned}}$

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